layered silicate nanocomposites

Polymer Testing 31 (2012) 345–354

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Property modelling

Predictive modeling of creep in polymer/layered silicate nanocomposites Ali Shokuhfar a, Abolfazl Zare-Shahabadi b, *, Ali-Asghar Atai c, Salman Ebrahimi-Nejad a, Mahdie Termeh a a

Advanced Materials and Nanotechnology Research Lab, Department of Mechanical Engineering, K.N. Toosi University of Technology, 19991-43344 Tehran, Iran Department of Mechanical Engineering, Yazd Branch, Islamic Azad University, Yazd, Iran c Department of Mechanical Engineering, University of Tehran, Tehran, Iran b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 November 2011 Accepted 23 December 2011

A predictive creep model is developed which uses the properties of matrix and reinforcement to predict the creep of polymer/layered silicate nanocomposites. Up to this point, primarily empirical creep models such as Findley and Burgers models have been used for creep of polymer/clay nanocomposites. The proposed creep model is based on the elastic-viscoelastic correspondence principle and a stiffness model of these nanocomposites. Also, the added stiffness of polymeric matrix due to the constraining effect of layered silicates on polymer chains in the nanocomposite is considered by a parameter termed constraint factor. The results of the proposed model show good agreement with experimental creep data for different clay contents, stresses and temperatures. Comparing the model predictions with experimental data, a logical relationship between the method of processing and the constraint factor is discovered which shows that in-situ polymerization can be more efficient for improving creep resistance of polymer/layered silicate nanocomposites relative to melt processing. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Polymer/clay nanocomposites Predictive modeling Creep resistance Processing method

1. Introduction Polymer/clay nanocomposites have attracted considerable attention from many researchers over the last few years because of the potentially higher properties that these nanocomposites can exhibit compared to conventional composites. Layered silicates or clay minerals are composed of very thin (w1 nm) platelets that have large surface areas and high aspect ratios. In addition, these platelets have very high stiffness (w178 GPa) compared to that of polymers (w3 GPa) [1]. Because of such stiffness and aspect ratios, these materials can be very efficient reinforcements, and in the last few years numerous studies have shown that minimal amounts of layered silicates can lead to significant enhancement of many mechanical and

* Corresponding author. Tel.: þ98 913 250 6474; fax: þ98 351 821 0670. E-mail addresses: [email protected], zare.shahabadi@ gmail.com (A. Zare-Shahabadi). 0142-9418/$ – see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2011.12.013

physical properties, including stiffness [2], strength [3], creep [4], flammability [5], permeability and gas barrier properties [6,7], and thermal stability [8,9]. Because of these attractive properties, polymer/clay nanocomposites are being used for a wide variety of applications, such as in transportation, construction, electronics and consumer products [10], and different types of polymeric materials including addition polymers [11–13], condensation polymers [14–16], biodegradable polymers [17,18] and asphalt binders [19,20] have been used as the matrix in these nanocomposites. The structures available for polymer/clay composites and nanocomposites can be divided into four categories: (a) conventional composite structures in which the silicate platelets are agglomerated (Fig. 1a), (b) intercalated nanocomposite structures, where some matrix molecules are inserted between individual silicate layers, but the layers remain parallel (Fig. 1b), (c) partially intercalated and exfoliated nanocomposites in which exfoliated layers and intercalated stacks are randomly distributed in the

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Fig. 1. Possible structures for polymer/layered silicate composites and nanocomposites: (a) conventional composite, (b) fully intercalated nanocomposite, (c) partially intercalated and exfoliated nanocomposite and (d) fully exfoliated nanocomposite [21].

matrix (Fig. 1c) and (d) fully exfoliated nanocomposite structures, in which the layered structure of the clay is disrupted, and the silicate layers are no longer close enough to interact with each other and the nanometric platelets are fully dispersed in the matrix (Fig. 1d). The complete dispersion and exfoliation of clay platelets inside the matrix leads to maximum improvements and is the ideal case. However, in most nanocomposites partial exfoliation and intercalation occurs and complete exfoliation is difficult to achieve [21]. The main methods of mass production of polymer/clay nanocomposites are melt processing [22–24] and in-situ polymerization [25–27]. In melt processing, ready molten polymer is used for production of nanocomposite, whereas in in-situ polymerization the clay platelets are dispersed in monomers and then the polymerization of monomers leads to formation of nanocomposite. In the in-situ polymerization method, an end-tethered polymer/clay nanocomposite hybrid is formed in which at least one end of the polymer chains is attached to the surface of layered silicates [28], whereas in the other methods the ends of polymer chains are free. Creep is one of important characteristics of thermoplastic polymers and thermoplastic polymer-based nanocomposites. This phenomenon controls dimensional stability of parts, especially when they are under stress for a long period of time. During creep in a material, stress remains constant and strain increases with time. The ratio,

SðtÞ ¼

s

(1)

3 c ðtÞ

is called the creep stiffness, where, 3 c and t are creep strain and time, respectively. In linearly viscoelastic materials, the creep stiffness is independent of stress level, whereas, in a nonlinear viscoelastic material it is a function of stress. Also, the creep stiffness is a function of temperature, and can be shifted between different temperatures using the time-temperature superposition principle [29]. Therefore, creep performance of a general viscoelastic material at stress s and temperature T can be designated by:

Sðt; s; TÞ ¼

s s; TÞ

3 c ðt;

(2)

Up to this point, common empirical models such as Burgers and Findley models have been used for creep in polymer/clay nanocomposites [4,30–32]. The empirical models are very general, and not specific for these nanocomposites. Each of these empirical models has a mathematical equation with some constants. For example, the formulation of the Findley model is: 3 ðtÞ

¼

30

þ mt n

(3)

where 3 0 is instantaneous strain, and m and n are material constants. These constants (3 0, m and n) must be determined from experimental creep data for any new material [33].

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Although, these empirical models can provide a time dependent mathematical equation for creep strain of a material, they cannot predict creep in a composite or nanocomposite using the properties of constituent materials. In this paper, a predictive creep model for polymer/ layered silicate nanocomposites will be proposed using the elastic-viscoelastic correspondence principle and a stiffness model developed previously [34]. Also, effect of layered silicate platelets on constraining polymer chains during creep will be studied using a parameter in this model termed “constraint factor”. 2. Modeling procedure 2.1. Brief summary of stiffness model In the development of the current creep model we need a stiffness model for polymer/layered silicate nanocomposites. The stiffness model used is from a previous paper by Zare-Shahabadi et al. [34].This model is based on the Halpin-Tsai composite model [35], and consideration of exfoliation ratio (the ratio of exfoliated filler content to the total filler content) re as a function volume fraction of filler ff helps this model be more accurate than other micromechanical models. The model suggests a linear exfoliation ratio function with negative slope:




re ¼ f ff ¼

8 <

re0  mff :0

ff 

re0 m

re0  ff m

(4)

where re0 and m are the initial exfoliation ratio at very low concentrations of filler and exfoliation reduction factor with volume fraction of filler. The model considers a three phase composite system including matrix, intercalated particles and exfoliated platelets. For calculation of nanocomposite modulus, first, the matrix and the exfoliated particles are assumed to make a new matrix, namely the exfoliated matrix, and its stiffness was calculated using Halpin-Tsai theory. Then, assuming an intercalated stack as an effective particle, the modulus of the effective particle is calculated by the rule of mixtures. Finally, by adding the effective intercalated particles to the exfoliated matrix using Halpin-Tsai theory, the total modulus of nanocomposite Ec is calculated. The formulation of the stiffness model is lengthy and, hence, it is shown briefly in this paper by the following function:


 Ec ¼ fstiffness Ef ; Em ; ff ; re

(5)

where Ef and Em are elastic modulus of filler and matrix, respectively. 2.2. Development of creep model In the model presented here, creep of the nanocomposite is considered as a decrease of stiffness of the polymeric matrix with time under stress, while the stiffness of the reinforcement is kept constant and unaffected by creep (Fig. 2). This model is close to the real case,

Fig. 2. Creep mechanism in the presented model is considered as reduction in creep stiffness of polymeric matrix in the nanocomposite.

because in this type of nanocomposite both polymer and ceramic components are under stress and at temperatures below 200  C at which creep tests on polymers and composites based on them take place. The ceramic component of the nanocomposite, i.e. layered silicates, has great resistance to creep due to high-strength ionic bonds, and hence show constant stiffness during the test. On the other hand, polymer component, which has weak bonds between its macromolecules, shows creep with time and its creep stiffness is time-varying. The assumptions in this model are as follows: 1- The matrix is considered homogeneous and isotropic. 2- The model does not consider the effects of aging and variation in nature and chemistry of the matrix. The viscoelastic aging effects on creep are also neglected. 3- The effects of intermediate phase are neglected. In other words, the interphase between the matrix and reinforcement is taken to be the same as the matrix. 4- The time-dependent creep stiffness of the matrix in all points of the nanocomposite is considered as a function of applied stress s (average stress) and not as a function of local stress. Therefore, in each applied stress s we consider the nonlinear viscoelastic matrix as a linear viscoelastic material with this creep stiffness. Based on the principle of elastic-viscoelastic correspondence [36], a stiffness model of the nanocomposite can be used in order to predict its creep stiffness. This principle states that if a linear viscoelastic body is under the action of constant load at time zero, its stress distribution is the same as that in an elastic body with the same geometry and under the same loading. Moreover, strains and displacements of the body are time-dependent and are obtained by substitution of the Young modulus with time-dependent creep stiffness into the governing equations of the elastic body. The present model takes the time variations of creep stiffness of the matrix Smexp ðt; s; TÞ, resulting from creep testing of the pure matrix at stress s and temperature T, and the constant stiffness of the reinforcement Ef as input and,

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using the stiffness model of polymer/layered silicate nanocomposites developed in reference [34], predicts the variations of creep stiffness of the nanocomposite, Snc ðt; s; TÞ at the same stress and temperature. This is done through substitution in the stiffness model of the Young’s modulus Em by the time varying creep stiffness of the matrix, Sm ðt; s; TÞ, and thus the creep stiffnessSnc ðt; s; TÞ of the nanocomposite is obtained as a function of time. Hence, the creep stiffness of the nanocomposite can be stated as


 Snc ðt; s; TÞ ¼ fstiffness Ef ; Sm ðt; s; TÞ; ff ; re

(6)

In the first stage of modeling, the creep stiffness of the nanocomposite is taken to be equal to that of the pure matrix, Smexp ðt; s; TÞ, i.e.

Sm ðt; s; TÞ ¼ Smexp ðt; s; TÞ

(7)

However, as seen later, sometimes with the advance of creep time, the creep strain obtained from this type of model gradually surpasses that from experiments. As a result, the creep stiffness of the matrix in the nanocomposite would be somewhat higher than that of the pure matrix. This can be justified by the fact that, compared to the pure polymeric matrix, the presence of the distributed nanolayers in the matrix puts additional constraints on the long chains and molecules of the polymer and prevents their movement. This phenomenon is termed here “added constraint stiffness effect”. Therefore, in the second stage of modeling, the creep stiffness of the matrix in the nanocomposite is taken to be higher than that of the pure matrix in order to take this constraining effect into account. It may be argued that this augmentation can simply be implemented through adding a positive quantity to the creep stiffness of the pure matrix, or by multiplying it by a positive factor.

First, the effect of the additive term is examined, and the creep stiffness of the matrix is taken to be

Sm ðt; s; TÞ ¼ Smexp ðt; s; TÞ þ Sc

(8)

in which Sc is the added stiffness due to constraint. This quantity is proportional to the volume percent of the reinforcement, since the more filler, the higher is the constraint on the polymer chains. On the other hand, as the creep strain goes up, it is clear that due to elongation of polymeric chains, their engagement with nanolayers increases. Hence, the added stiffness is also proportional to the creep strain of the matrix. Therefore, the following mathematical form is suggested for the added stiffness

Sc ¼ kc

ff 3 mexp ðt;

(9)

s; TÞ

and using Eq. (2) it can be written

Sc ¼ kc

sff

(10)

Smexp ðt; s; TÞ

and Eq. (8) becomes

Sm ðt; s; TÞ ¼ Smexp ðt; s; TÞ þ kc

sff

Smexp ðt; s; TÞ

(11)

However, this relation must satisfy a specific condition; at the beginning of creep and when the time is very close to zero, the creep stiffness of the matrix in the nanocomposite must be equal to that of the pure matrix, i.e.

Sm ð0; s; TÞ ¼ Smexp ð0; s; TÞ Taking t ¼ 0 in Eq. (11) gives

Fig. 3. Effect of exfoliation ratio on the results of the creep model in different filler contents, and no added constraint stiffness effect (kc ¼ 0).

(12)

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Sm ð0; s; TÞ ¼ Smexp ð0; s; TÞ þ kc

sff

Smexp ð0; s; TÞ

sSmexp ð0; s; TÞ (13)

So, this mathematical form does not satisfy the above condition and hence is not valid. Therefore, to remedy this issue, another mathematical relation must be suggested. The following form is now examined

Sm ðt; s; TÞ ¼ Smexp ðt; s; TÞ$



 s; TÞ ðkc $ff Þ 3 mexp ð0; s; TÞ 3 mexp ðt;

  f Smexp ð0; s; TÞ ðkc $ f Þ Smexp ðt; s; TÞ

exfoliated (33% and 66% exfoliation), and fully exfoliated nanocomposites with different filler contents. As is seen, intercalated nanocomposites show more resistance to creep than pure polymer, but when exfoliation is increased to 33%, the creep resistance of nanocomposite is nearly doubled. The trend of increase of creep resistance with the increase of exfoliation ratio continues although at a lower rate. As expected, with increase of exfoliation, the effect of adding more filler becomes greater, and creep resistance increases.

(14) 3.2. Effect of constraint factor

The motivation behind this suggestion is that this simple form with multiplicative type of augmentation is directly proportional to the creep strain and volume percentage of the reinforcement and it also satisfies the necessary condition at time zero. Using Eq. (2) it can be written

Sm ðt; s; TÞ ¼ Smexp ðt; s; TÞ$

349

(15)

in which Smexp ð0; s; TÞ is the initially measured creep stiffness at the instant just after loading at time zero. kc is termed the constraint factor which is a non-negative real number. It is an indication of increase of creep stiffness of matrix in nanocomposite with the increase in creep strain. The constraint factor can be dependent on factors such as polymer type, length of polymeric chains (molecular weight of the polymer), type of modification on the surface of nanoclay and production method of the nanocomposite. By substituting Eq. (15) into Eq. (6), the creep stiffness of the nanocomposite is as follows

Fig. 4 shows the effect of constraint factor on predictions of creep model for 2, 4 and 6 percent of filler content. As seen in this figure, with increasing constraint factor, the predicted creep and the slope of creep diagram decreases. Also, as expected, with increasing filler content the effect of constraint factor on the creep diagrams increases. 3.3. Model predictions and experimental results 3.3.1. Nylon 6/clay nanocomposites Fig. 5(a) shows the forecasts of the model, regardless of the effect of the added constraint stiffness and with a zero constraint factor for nylon-based nanocomposites with 2 and 5% nanoclay, compared with the experimental results of Nakazato et al. [31]. These nanocomposites have been synthesized using in-situ polymerization. As can be seen in Fig. 5(a), predictions of the model without considering the effect of added constraint stiffness go beyond the experimental results as the time increases. In other words, over time, the estimated creep stiffness will become less than the real stiffness achieved in the experimental results.

 f      Smexp ð0; s; TÞ ðkc : f Þ ; ff ; re Snc ðt; s; TÞ ¼ fstiffness Ef ; Smexp ðt; s; TÞ$ Smexp ðt; s; TÞ

(16)

Finally, using Eq. (2), the creep strain of the nanocomposite can be written as

εnc ðt; s; TÞ ¼

s

     4  Smexp ð0; s; TÞ ðkc $ f Þ ; 4 f ; re fstiffness Ef ; Smexp ðt; s; TÞ$ Smexp ðt; s; TÞ

The constraint factor kc in the model is a parameter which must be identified with experimental curves using the leastsquares method. The programming and computations for the current model were done with MATLAB software. 3. Results and discussion 3.1. Effect of filler content and exfoliation ratio Fig. 3 shows the predictions of the model without constraint factor (kc ¼ 0) for intercalated, partially

(17)

This can be justified by the fact that when the nanocomposite is subjected to creep, the silicate layers constrain the polymer chains and hence, impede their ease of movement. As a result, the creep stiffness of the matrix increases compared to that obtained without the presence of nanoparticles (experimental results of pure polymer). Fig. 5(b) shows the predictions of the model considering the effect of the added constraint stiffness. As can be seen in Fig. 5(b), considering the effect of added constraint stiffness has significantly improved the model

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of Yang et al. [4] are used. The creep test has only been conducted for a single weight percent of nanoclay (1.6 wt%) but the results can be used to investigate the performance of the model at different temperatures and stresses. Fig. 6 illustrates the predictions of the model at temperatures of 50 and 80  Celsius and for stress values of 30 and 40 MPa. As can be seen, the creep model with identical stiffness functions and zero constraint factor is in good agreement with experimental curves at different temperatures and stresses. 3.3.3. Starch-polycaprolactone/clay nanocomposites Fig. 7 shows the predictions of the creep model compared with the experimental results for starchpolycaprolactone/clay nanocomposite extracted from the work of Perez et al. [32]. As can be seen, the model with the same stiffness function and a zero constraint factor is in good agreement with experimental results for different percentages of clay. Parameters used for predictions of the model in figs 5–7 are listed in Table 1. 3.4. Comparison of constraint factors in the experimental results

Fig. 4. Effect of constraint factor on the results of the creep model with (a) 2%, (b) 4% and (c) 6% nanoclay content of the nanocomposite.

predictions compared to those of Fig. 5(a). The maximum deviation of the predictions is reduced from 25% in Fig. 5(a) to 3% in Fig. 5(b) using a non-zero constraint factor. 3.3.2. Polyamide 66/clay nanocomposites To check the model results for nanocomposites based on polyamide 66, in this section, the experimental results

According to the logic expressed in this model, it can be said that in nanocomposites with a zero constraint factor, the polymer matrix shows the same creep stiffness and creep resistance as that shown by the pure matrix in creep experiments. Non-zero constraint factors indicate the effect of the nanolayers in preventing the movement of polymer chains during creep of the nanocomposite. In other words, constraint factor is a measure of how effectively the nanoparticles constrain the polymer chains. As seen in Section 3.3, in nanocomposites based on polyamide 66 and starch-polycaprolactone, the constraint factor is zero, and only in the case of nylon 6-based nanocomposite is it non-zero. Polyamide 66 and nylon 6 are very similar in terms of the chemical composition of chains and are both from the polyamide family. More detailed investigation reveals that the main difference in the studied nanocomposites is in their production methods; nanocomposites based on polyamide 66 and starch-polycaprolactone are produced through melt processing methods, whereas the nylon 6-based nanocomposite is produced through in-situ polymerization. So, it seems that compared with the melt processing method, in the in-situ polymerization method, the relative situation of polymer chains and silicate nanolayers is such that it engages the polymer chains with the clay nanolayers and, therefore, during creep the motion of the polymer chains faces more resistance. The interesting point is that in nanocomposites produced by in-situ polymerization the polymer chains are tethered to the surface via strong ionic interactions between the silicate layer and the polymer end [28], whereas, in the melt processing method, molten polymer is mixed with nanoclay under shearing stresses of extrusion and the ends of polymer chains are free when

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Fig. 5. Comparison of predictions of the creep model and experimental results for Nylon 6/clay nanocomposites [31] in different clay contents (a) with zero constraint factor (b) with non-zero constraint factor (kc ¼ 25).

the nanocomposite is formed. Also, in nanocomposites produced by melt processing only weak van der Waals and hydrogen bonds between polymer chains and organic surface modification of silicate platelets exist [37]. These weak bonds also exist between polymer chains in the bulk polymer. Therefore, the time variations of creep stiffness in bulk polymer and time variations of creep stiffness in the matrix of nanocomposite are very close together.

In addition, it should be noted that the melt processing method is based on the flow of molten polymer relative to the layered silicates, which causes the nanolayers and the polymer chains to be situated in a position to be able to move easily relative to each other, after the formation of nanocomposites. Another result which can be proposed using this model is that in-situ polymerization is a better method for increasing creep resistance compared to melt processing.

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Fig. 6. Comparison of predictions of the creep model and experimental results for Polyamide 66/clay nanocomposites [4] in different stresses and temperatures with zero constraint factor.

Fig. 7. Comparison of predictions of the creep model and experimental results for Starch-polycaprolactone/clay nanocomposites [32] in different clay contents with zero constraint factor.

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Table 1 Parameters used to calculate the predictions of the model in Figs. 5–7. Polymer

Ef (GPa)

Matrix density (gr/cm3)

Fiber density (gr/cm3)

Platelet thickness (nm)

Interlayer spacing (nm)

Silicate platelet length (nm)

Nylon 6 Polyamid 66 Starch/polycaprolactone

178 178 178

1.16 1.14 1

2.44 2.44 2.44

1.4 1.4 1.4

4.8 4.8 4.8

147 170 250

4. Conclusions A new model for the creep of polymer/layered silicate nanocomposites is developed in this paper based on a micromechanical stiffness model and the principle of elastic-viscoelastic correspondence. This model uses the characteristics of the polymeric matrix and the reinforcement silicate layers to predict the nanocomposite’s creep behavior. To the best of the authors’ knowledge, this model is the first predictive model for studying the creep of polymer/layered silicate nanocomposites. The increase in creep resistance of polymer/layered silicate nanocomposite is explained using two mechanisms: 1. Increased creep stiffness of the nanocomposite due to stress transfer from the polymer matrix to the nanolayers: This increase in the creep stiffness is incorporated through the micromechanical stiffness model and the principle of elastic-viscoelastic correspondence. 2. The effect of constraint factor due to the restrictions of movement of polymer chains by the nanolayers: this creep stiffness with non-zero coefficient constraint factor is added to the creep stiffness achieved through the micromechanical stiffness model. Predictions of the model were compared with experimental results of three types of nanocomposite with different matrices. For each of the studied nanocomposites, the predictive creep model is able to provide predictions according to different filler percentages or different temperature and stress conditions, with the same stiffness function and similar constraint factors, in each case. By comparing predictions of the creep model with empirical data, some interesting results about the relationship between constraint factor (which represents how well the polymer chains movement is prevented by the nanolayers) and the nanocomposite manufacturing method was obtained; Compared to melt processing, insitu polymerization seems to be a better method to prevent the movement of the polymer chains during the creep of the produced nanocomposite. These findings that have been confirmed through modeling are reasonably well supported because:  Melt processing is based on the flow of molten polymer which causes the nanolayer and the polymer chains to be situated in a position to be able to move easily compared to each other after the formation of the nanocomposites and, therefore, the constraint factor of nanocomposites produced through the melt processing method is zero or close to zero.

 On the other hand, in the in-situ polymerization method, polymerization and the formation of nanocomposite are done simultaneously and, more importantly, at least one end of the polymer chains are attached to the surface of the nanolayers and thus the constraint factor of nanocomposites produced through in-situ polymerization method is non-zero.

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